Finite Group Databasebeta

← #33293149

Subgroups — #33293149

Order 18688.

1854 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618218C212yesmaximal
2#22617126C214yesmaximal, complemented
3#22616836C214yesmaximal, complemented
4#22616589C214yesmaximal, complemented
5#22616397C214yesmaximal, complemented
6#22616045C211yesmaximal, characteristic, complemented
7#22615901C211yesmaximal, characteristic, complemented
8#22614171C214yesmaximal
9#22614066C214yesmaximal, complemented
10#22612664C212yesmaximal, complemented
11C2×#380×C73C211yesmaximal, characteristic
12C2×#14556584C2214yes—
13C2×#14556581C2214yescomplemented
14C2×#14556570C2212yes—
15C2×#14556570C2212yes—
16C2×#14556570C2212yes—
17#17988529C2214yescomplemented
18#17988525C2214yescomplemented
19#17988521C2214yescomplemented
20#17988520C2214yescomplemented
21#17988519C2214yescomplemented
22#17988518C2214yescomplemented
23#17988517C2214yescomplemented
24#17988515C2214yes—
25#17988515C2214yes—
26#17988512C2214yes—
27#17988510C2214yes—
28#17988508C2214yescomplemented
29#17988491C2214yescomplemented
30#17988484C2214yescomplemented
31#17988482C2214yescomplemented
32#17988481C2214yes—
33#17988480C2214yes—
34#17988475C2214yes—
35#17988474C2214yescomplemented
36#17988473C2214yescomplemented
37#17988470C2214yescomplemented
38#17988468C2214yescomplemented
39#17988460C2214yes—
40#17988460C2214yes—
41#17988460C2214yes—
42#17988460C2214yes—
43#17988460C2214yes—
44#17988447C2212yes—
45#17988447C2212yes—
46#17988447C2214yes—
47#17988447C2212yes—
48#17988443C2214yes—
49C4×C22⋊Dic73C2214yes—
50C24×C4×C73C2211yescharacteristic
51C22×C4⋊C4×C73C2212yes—
52C22×C22⋊C4×C73C2212yes—
53#380×C73C2214yes—
54#380×C73C2214yes—
55C2×#95×C73C2211yescharacteristic
56C2×#95×C73C2211yescharacteristic
57C22×Dic146C2312yes—
58#14556584—48no—
59#14556581—48no—
60#14556570—28no—
61#14556570—44no—
62#14556570—44no—
63C2×C4⋊Dic73C2312yes—
64C2×C4⋊Dic73C2312yes—
65C2×C4⋊Dic73C2312yes—
66C2×C4⋊Dic73C2314yes—
67C2×C4⋊Dic73C2314yes—
68C2×#11924007C2314yes—
69C2×#11924007C2314yes—
70C2×#11924007C2314yes—
71C2×#11924007C2314yes—
72C2×#11924007C2314yes—
73C2×#11924007C2314yes—
74C2×#11924007C2314yes—
75C2×C22⋊Dic73C2314yes—
76C2×C22⋊Dic73C2314yes—
77C2×C22⋊Dic73C2314yes—
78C2×C22⋊Dic73C2314yes—
79C2×C22⋊Dic73C2314yes—
80C2×C22⋊Dic73C2314yes—
81C2×C4×Dic73C2314yes—
82C2×C4×Dic73C2314yes—
83#14556498C2314yes—
84#14556498C2314yes—
85#14556498C2314yes—
86#14556498C2314yes—
87#14556498C2314yes—
88#14556498C2314yes—
89#14556498C2314yes—
90#14556498C2314yes—
91#14556498C2314yes—
92#14556498C2314yes—
93#14556498C2314yes—
94C25×C73C2311yescharacteristic
95C23×C4×C73—24no—
96C23×C4×C73—44no—
97C23×C4×C73C2311yescharacteristic
98C23×C4×C73C2312yes—
99C23×C4×C73C2312yes—
100C23×C4×C73C2312yes—
101C23×C4×C73C2312yes—
102C23×C4×C73C2311yescharacteristic
103C2×C4⋊C4×C73—44no—
104C2×C4⋊C4×C73C2314yes—
105C2×C4⋊C4×C73C2314yes—
106C2×C22⋊C4×C73—44no—
107C2×C22⋊C4×C73C2314yes—
108C2×C22⋊C4×C73C2314yes—
109#95×C73C2314yes—
110#95×C73C2314yes—
111C2×Dic146—24no—
112C2×Dic146—42no—
113C2×Dic146—42no—
114C22×Dic73C2414yes—
115C22×Dic73C2414yes—
116C22×Dic73C2414yes—
117C22×Dic73C2414yes—
118C4⋊Dic73—24no—
119C4⋊Dic73—42no—
120C4⋊Dic73—42no—
121C4⋊Dic73—44no—
122C4⋊Dic73—44no—
123#11924007—28no—
124#11924007—28no—
125#11924007—28no—
126#11924007—44no—
127#11924007—44no—
128#11924007—44no—
129#11924007—44no—
130C22⋊Dic73—28no—
131C22⋊Dic73—28no—
132C22⋊Dic73—28no—
133C22⋊Dic73—28no—
134C22⋊Dic73—44no—
135C22⋊Dic73—44no—
136C4×Dic73—44no—
137C4×Dic73—44no—
138C24×C73—22no—
139C24×C73—22no—
140C24×C73—22no—
141C24×C73—24no—
142C24×C73—42no—
143C24×C73C2411yescharacteristic
144C24×C73C2411yescharacteristic
145C24×C73C2411yescharacteristic
146C22×C4×C73—22no—
147C22×C4×C73—24no—
148C22×C4×C73—24no—
149C22×C4×C73—24no—
150C22×C4×C73—24no—
151C22×C4×C73—24no—
152C22×C4×C73—24no—
153C22×C4×C73—24no—
154C22×C4×C73—42no—
155C22×C4×C73—42no—
156C22×C4×C73—42no—
157C22×C4×C73—42no—
158C22×C4×C73—42no—
159C22×C4×C73—42no—
160C22×C4×C73—42no—
161C22×C4×C73—416no—
162C22×C4×C73#4114yes—
163C22×C4×C73C2414yes—
164C22×C4×C73C2414yes—
165C22×C4×C73C2412yes—
166C22×C4×C73C2412yes—
167C4⋊C4×C73—28no—
168C4⋊C4×C73—44no—
169C22⋊C4×C73—28no—
170C22⋊C4×C73—44no—
171C23×C73—22no—
172C23×C73—21no—
173C23×C73—21no—
174C23×C73—28no—
175C23×C73—21no—
176C23×C73—21no—
177C23×C73—21no—
178C23×C73—22no—
179C23×C73—22no—
180C23×C73—22no—
181C23×C73—22no—
182C23×C73—21no—
183C23×C73—48no—
184C23×C73—48no—
185C23×C73—44no—
186C23×C73—44no—
187C23×C73C2×#4112yes—
188C23×C73H(5,𝔽2)12yes—
189C23×C732−1+414yes—
190C23×C732−1+412yes—
191C23×C73C2511yescharacteristic
192C2×C4×C73—28no—
193C2×C4×C73—22no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—24no—
198C2×C4×C73—24no—
199C2×C4×C73—24no—
200C2×C4×C73—24no—
201C2×C4×C73—24no—
202C2×C4×C73—24no—
203C2×C4×C73—44no—
204C2×C4×C73—48no—
205C2×C4×C73—48no—
206C2×C4×C73—84no—
207C2×C4×C73—84no—
208C2×C4×C73C2×#4114yes—
209C2×C4×C73H(5,𝔽2)14yes—
210C2×Dic73—24no—
211C2×Dic73—24no—
212C2×Dic73—24no—
213C2×Dic73—24no—
214C2×Dic73—24no—
215C2×Dic73—24no—
216C2×Dic73—24no—
217C2×Dic73—24no—
218C2×Dic73—24no—
219C2×Dic73—24no—
220C2×Dic73—24no—
221C2×Dic73—24no—
222Dic146—84no—
223C22×C73—22no—
224C22×C73—22no—
225C22×C73—24no—
226C22×C73—24no—
227C22×C73—22no—
228C22×C73—22no—
229C22×C73—24no—
230C22×C73—24no—
231C22×C73—416no—
232C22×C73—41no—
233C22×C73—41no—
234C22×C73—41no—
235C22×C73—41no—
236C22×C73—42no—
237C22×C73—42no—
238C22×C73#54114yes—
239C22×C73C22×#4111yescharacteristic
240C22×C73C2×H(5,𝔽2)12yes—
241C22×C73C2×H(5,𝔽2)11yescharacteristic
242C22×C73C2×2−1+412yes—
243C22×C73C2×2−1+411yescharacteristic
244C4×C73—24no—
245C4×C73—24no—
246C4×C73—44no—
247C4×C73—44no—
248Dic73—44no—
249Dic73—44no—
250Dic73—44no—
251Dic73—44no—
252#36728—731nomaximal, Sylow
253C2×C73—24no—
254C2×C73—42no—
255C2×C73—42no—
256C2×C73C2×#54111yescharacteristic
257C2×C73#349011yescharacteristic
258C2×C73#349211yescharacteristic
259C2×C73#353612yes—
260C2×C73#353812yes—
261#2848—732no—
262#2747—734no—
263#2722—734no—
264#2704—734no—
265#2690—734no—
266#2659—731no—
267#2646—731no—
268#2480—734no—
269#2473—734no—
270#2329—732no—
271C2×#380D7311yescharacteristic
272C73#3672811yescharacteristic, complemented, Sylow
273C24×C4C2×D7311yescharacteristic
274C2×#126—734no—
275C2×#125—734no—
276C2×C22⋊Q8—732no—
277C2×C22⋊Q8—732no—
278C2×C22⋊Q8—732no—
279C22×C4⋊C4C2×D7312yes—
280C22×C22⋊C4C2×D7312yes—
281#400—734no—
282#400—734no—
283#398—734no—
284#398—734no—
285#398—734no—
286#398—734no—
287#398—734no—
288#397—734no—
289#397—734no—
290#397—734no—
291#396—734no—
292#395—734no—
293#391—734no—
294#389—734no—
295#389—734no—
296#388—734no—
297#388—734no—
298#388—734no—
299#387—734no—
300#387—734no—
301#387—734no—
302#387—734no—
303#385—734no—
304#385—734no—
305#385—734no—
306#385—734no—
307#385—734no—
308#380—732no—
309#380—732no—
310#380—732no—
311#380—734no—
312#380C2×D7314yes—
313#380C2×D7314yes—
314#379—734no—
315C4×C22⋊C4—734no—
316C2×#95C2×D7311yescharacteristic
317C2×#95C2×D7311yescharacteristic
318C25C22×D7311yescharacteristic
319C22×Q8—732no—
320C23×C4—24no—
321C23×C4—44no—
322C23×C4C22×D7311yescharacteristic
323C23×C4C22×D7312yes—
324C23×C4C22×D7312yes—
325C23×C4C22×D7312yes—
326C23×C4C22×D7312yes—
327C23×C4C22×D7311yescharacteristic
328#126—2928no—
329#125—2928no—
330C22⋊Q8—2924no—
331C22⋊Q8—2924no—
332C22⋊Q8—1468no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—732no—
335C2×C4⋊C4—732no—
336C2×C4⋊C4—734no—
337C2×C4⋊C4—734no—
338C2×C4⋊C4—734no—
339C2×C4⋊C4—734no—
340C2×C4⋊C4—734no—
341C2×C4⋊C4—734no—
342C2×C4⋊C4—734no—
343C2×C4⋊C4—734no—
344C2×C4⋊C4—732no—
345C2×C4⋊C4—734no—
346C2×C4⋊C4C22×D7314yes—
347C2×C4⋊C4C22×D7314yes—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—734no—
350C2×C22⋊C4—734no—
351C2×C22⋊C4—734no—
352C2×C22⋊C4—734no—
353C2×C22⋊C4—734no—
354C2×C22⋊C4—734no—
355C2×C22⋊C4C22×D7314yes—
356C2×C22⋊C4C22×D7314yes—
357C2×C42—734no—
358C2×C42—734no—
359#95—734no—
360#95—734no—
361#95—734no—
362#95—734no—
363#95—734no—
364#95—734no—
365#95—734no—
366#95—734no—
367#95—734no—
368#95—734no—
369#95—734no—
370#95C22×D7314yes—
371#95C22×D7314yes—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—42no—
377C24C23×D7311yescharacteristic
378C24C23×D7311yescharacteristic
379C24C23×D7311yescharacteristic
380C2×Q8—2922no—
381C2×Q8—2922no—
382C2×Q8—1464no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—416no—
399C22×C4—734no—
400C22×C4—734no—
401C22×C4—734no—
402C22×C4—734no—
403C22×C4#1192402814yes—
404C22×C4C23×D7314yes—
405C22×C4C23×D7314yes—
406C22×C4C23×D7312yes—
407C22×C4C23×D7312yes—
408C4⋊C4—28no—
409C4⋊C4—44no—
410C4⋊C4—2924no—
411C4⋊C4—2924no—
412C4⋊C4—2924no—
413C4⋊C4—2924no—
414C4⋊C4—2922no—
415C4⋊C4—2922no—
416C4⋊C4—2924no—
417C4⋊C4—2924no—
418C4⋊C4—1468no—
419C4⋊C4—1464no—
420C4⋊C4—1468no—
421C4⋊C4—1468no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—2924no—
425C22⋊C4—2924no—
426C22⋊C4—1468no—
427C22⋊C4—1468no—
428C22⋊C4—1468no—
429C22⋊C4—1468no—
430C42—2924no—
431C42—2924no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—28no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—21no—
444C23—48no—
445C23—48no—
446C23—44no—
447C23—44no—
448C23C2×#1192402812yes—
449C23#1455663612yes—
450C23#1455663814yes—
451C23#1455663912yes—
452C23C24×D7311yescharacteristic
453Q8—5844no—
454C2×C4—28no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—1464no—
471C2×C4—1464no—
472C2×C4—1464no—
473C2×C4—1464no—
474C2×C4—1464no—
475C2×C4—1464no—
476C2×C4—1464no—
477C2×C4—1464no—
478C2×C4—1464no—
479C2×C4—1464no—
480C2×C4—1464no—
481C2×C4—1464no—
482C2×C4C2×#1192402814yes—
483C2×C4#1455663614yes—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—24no—
488C22—22no—
489C22—22no—
490C22—24no—
491C22—24no—
492C22—416no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—42no—
498C22—42no—
499C22#1798919914yes—
500C22C22×#1192402811yescharacteristic
501C22C2×#1455663612yes—
502C22C2×#1455663611yescharacteristic
503C22C2×#1455663812yes—
504C22C2×#1455663911yescharacteristic
505C4—24no—
506C4—24no—
507C4—44no—
508C4—44no—
509C4—2924no—
510C4—2924no—
511C4—2924no—
512C4—2924no—
513C2—24no—
514C2—42no—
515C2—42no—
516C2C2×#1798919911yescharacteristic
517C2#2262362011yescharacteristic
518C2#2262364011yescharacteristic
519C2#2262415312yes—
520C2#2262417612yes—
subgroupG/H
center, FrattiniC23C24×D73
commutator, socle, cosocleC23×C73C25
FittingC2×#380×C73C2
prime radicalC25×C73C23
hypercenterC2×#380D73
hypocenterC73#36728
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#380

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22617126C2semidirect
3. #22616836C2semidirect
4. #22616589C2semidirect
5. #22616397C2semidirect
6. #22616045C2semidirect
7. #22615901C2semidirect
9. #22614066C2semidirect
10. #22612664C2semidirect
13. C2×#14556581C22semidirect
17. #17988529C22semidirect
18. #17988525C22semidirect
19. #17988521C22semidirect
20. #17988520C22semidirect
21. #17988519C22semidirect
22. #17988518C22semidirect
23. #17988517C22semidirect
28. #17988508C22semidirect
29. #17988491C22semidirect
30. #17988484C22semidirect
31. #17988482C22semidirect
35. #17988474C22semidirect
36. #17988473C22semidirect
37. #17988470C22semidirect
38. #17988468C22semidirect
272. C73#36728semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226182182
#226171262
#226168362
#226165892
#226163972
#226160452
#226159012
#226141712
#226140662
#226126642
C2×#380×C732
#3672873